On the Martingale Property of Certain Local Martingales

Abstract

The stochastic exponential $Z_t=\exp \{M_t-M_0-(1/2)<M,M{>}_t\}$ of a continuous local martingale $M$ is itself a continuous local martingale. We give a necessary and sufficient condition for the process $Z$ to be a true martingale in the case where $M_t={\int }_0^{t}b(Y_u)d{W}_u$ and $Y$ is a one-dimensional diffusion driven by a Brownian motion $W$. Furthermore, we provide a necessary and sufficient condition for $Z$ to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function $b$ and the drift and diffusion coefficients of $Y$. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.

Appendix on local time of diffusions added; 27 pages, 1 figure; to appear in PTRF

References in corpus (1)

• Analysis of continuous strict local martingales via h-transforms

Cited by in corpus (1)

• When a Stochastic Exponential is a True Martingale. Extension of a Method of Bene^s